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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 76050bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76050.bs6 | 76050bc1 | \([1, -1, 0, 569583, -66404259]\) | \(371694959/249600\) | \(-13723100667900000000\) | \([2]\) | \(2064384\) | \(2.3609\) | \(\Gamma_0(N)\)-optimal |
| 76050.bs5 | 76050bc2 | \([1, -1, 0, -2472417, -550082259]\) | \(30400540561/15210000\) | \(836251446950156250000\) | \([2, 2]\) | \(4128768\) | \(2.7074\) | |
| 76050.bs3 | 76050bc3 | \([1, -1, 0, -21484917, 37950230241]\) | \(19948814692561/231344100\) | \(12719384508111876562500\) | \([2, 2]\) | \(8257536\) | \(3.0540\) | |
| 76050.bs2 | 76050bc4 | \([1, -1, 0, -32131917, -70042290759]\) | \(66730743078481/60937500\) | \(3350366373999023437500\) | \([2]\) | \(8257536\) | \(3.0540\) | |
| 76050.bs4 | 76050bc5 | \([1, -1, 0, -4373667, 96727373991]\) | \(-168288035761/73415764890\) | \(-4036426010402036738906250\) | \([2]\) | \(16515072\) | \(3.4006\) | |
| 76050.bs1 | 76050bc6 | \([1, -1, 0, -342796167, 2442964936491]\) | \(81025909800741361/11088090\) | \(609627304826663906250\) | \([2]\) | \(16515072\) | \(3.4006\) |
Rank
sage: E.rank()
The elliptic curves in class 76050bc have rank \(0\).
Complex multiplication
The elliptic curves in class 76050bc do not have complex multiplication.Modular form 76050.2.a.bc
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
